Skip to main content
SpringerLink
Log in

Existence Theorems for Bilevel Problem with Applications to Mathematical Program with Equilibrium Constraint and Semi-Infinite Problem

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we establish existence theorems for bilevel problems with fixed-point constraints and bilevel problems without fixed-point constraint. The aim of this paper is to investigate under which conditions the existence of a feasible point of a bilevel problem can be assumed in advance and under which conditions there exist minimizers for this type of problems. From this, we establish existence theorems for mathematical programs with equilibrium constraints and semi-infinite problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lin, L.J., Hsu, H.W.: Existence theorems of systems of vector quasi-equilibrium problems and mathematical programs with equilibrium constraints. J. Glob. Optim. 37, 195–213 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Birbil, S., Bouza, G., Frenk, J.B.G., Still, G.: Equilibrium constrained optimization problems. Eur. J. Oper. Res. 169, 1108–1127 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  4. Lin, L.J., Still, G.: Mathematical programs with equilibrium constraints: the structure of the feasible set and the existence of feasible points. Optimization 55, 205–219 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Stein, O.: Bilevel Strategies in Semi-Infinite Programming. Kluwer Academic, Dordrecht (2003)

    Google Scholar 

  6. Stein, O., Still, G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142, 444–462 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lin, L.J.: Mathematical programming with systems of equilibrium constraints. J. Glob. Optim. 37, 275–286 (2007)

    Article  MATH  Google Scholar 

  8. Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1994)

    Google Scholar 

  9. Lin, L.J., Yu, Z.T.: On some equilibrium problems for multivalued maps. J. Comput. Appl. Math. 129, 171–183 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tan, N.X.: Quasi-variational inequalities in topological linear locally convex Hausdorff spaces. Math. Nachr. 122, 231–245 (1985)

    Article  MathSciNet  Google Scholar 

  11. Himmelberg, C.J.: Fixed point of compact multifunctions. J. Math. Anal. Appl. 38, 205–207 (1972)

    Article  MathSciNet  Google Scholar 

  12. Fan, K.: Fixed point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. U.S.A. 38, 121–126 (1952)

    Article  MATH  Google Scholar 

  13. Lin, L.J., Ansari, Q.H.: Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296, 445–472 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Changhua University of Education, Changhua, Taiwan, ROC

    L. J. Lin

Authors
  1. L. J. Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. J. Lin.

Additional information

Communicated by J.P. Crouzeix.

This research was supported by the National Science Council of the Republic of China. The author thanks the referees for suggestions and comments leading to the present form of the paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, L.J. Existence Theorems for Bilevel Problem with Applications to Mathematical Program with Equilibrium Constraint and Semi-Infinite Problem. J Optim Theory Appl 137, 27–40 (2008). https://doi.org/10.1007/s10957-007-9283-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-007-9283-0

Keywords

Search

Navigation